Why do students need to understand this mathematics?

It is important that students understand the mathematics taught in this lesson for many reasons.  This mathematics will be applied in the future of the students, as well as in future mathematics courses.  This is an important component to mathematics and is important to forming complete mathematical understanding.  It is important for the teacher not to sugarcoat the lessons about measurements; studies have found that the "current practices of giving students squares as units may lead to apparent procedural competence but fail to challenge students' preconceptions about what makes a unit suitable" (National Council of Teachers of Mathematics, 2003, p. 185).  Although I do make reference to units in some portions of the unit, I do make sure that I am using measurements.  It is important for students to understand how to measure objects and how to convert between the various components of the customary and metric systems.  If students are going to be successful in their future, they must learn about finding the perimeter and circumference of objects, as well as the areas of figures.  These are concepts that will follow them throughout the rest of their lives!

Why is it appropriate for this course and/or grade level?

At this level, students have already been exposed to some measurement applications, as well to the ideas of perimeter, area, and circumference.  This is the first time, though, that students are really learning the concepts and applications of the concepts.  One benefit to this unit is that students utilize skills that have been previously learned.  This may be beneficial because the students will already have an idea of what is being discussed.  The ideas presented are not as foreign since the students have had some previous exposure to the concepts that are introduced.  This unit is approriate for this level because students are at the point where they need to understand measurement and the components that correspond to measurement.  Students must have an understanding about conversions, as well as area, perimeter, and circumference.  These are ideas that students must know because they are not only essential in the mathematics classroom, but also in the real world.

Why is it appropriate where this unit is situated in the curriculum?

Students will have already learned about various polygons; thus, they will understand how the shapes are similar and different.  Therefore, they will have an understanding of how the areas of the polygons relate.  Because of where thie unit is situated in the curriculum, the students are preparing to learn more about triangles and geometric concepts that utilize knowledge based on measurement.  For example, one of the next lessons relates to the Pythagorean Theorem.  Students can use the knowledge from this lesson an apply it to lessons that will be taught in the future.  The placement of this unit in the curriculum prepares students for upcoming mathematical concepts and ideas that will be presented.  It is important that students have the background from this unit to prepare them for the information that will be taught next.

How does this unit connect to the NCTM PSSM and to the NCSCOS?

In the NCTM PSSM, it is stated that all students in grades 6-8 should understand both metric and customary systems of measurement, understand relationships among units and convert from one unit to another within the same system, and understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.  Although not all of these concepts are covered in my unit, the majority of the expectations of the NCTM PSSM are covered in this unit.  Students are exposed to the expectations of NCTM and are hopefully learning about measurement.  In addition to measurement, students are exposed to communication and connections as well.  Students have to communicate their mathematical thinking coherently and clearly to peers, teachers, and others through their math journals, as well as through some of the activities that they participate in during class.  Students must communicate during class, especially if they are called on individually to answer a question.  If the students are communicating, though, it is the job of the teacher to make sure that the student is using the language of matehmatics correctly and precisely.  We want to make sure that our students are learning the correct way to communicate mathematics to others.  Students are also making connections between different mathematical concepts in this unit.  This is apparent in the lesson where students are finding the area of a circle.  They use knowledge learned about finding circumference to figure out the formula for finding the area of a circle.  Thus, in this unit alone, students are understanding how mathematical ideas interconnect and build on one another to produce a coherent whole.  In the problem solving lesson, as well as some of their homework problems, students recognize and apply mathematics in contexts outside of mathematics.  Students are using knowledge outside of the mathematics classroom and are applying it inside the mathemaics classroom.  In the NCSCOS, students are expected to
demonstrate an understanding and use of the properties and relationships in geometry, and standard units of metric and customary measurement.  This is one entire competency goal in itself; thus, students must be able to do anything that falls under this goal.

What special features are included in the unit and why are they valuable learning experiences?

There are many features in this unit that I think are of particular importance for students learning mathematics.  First of all, the unit incorporates a lesson on problem solving and real world applications.  This is an important component of the lesson because students see that why they need to learn the mathematical formulas and ideas presented in this unit.  All of the components of this lesson are used in the real world, particularly the lessons on the customary and metric measurement systems.  It is important, though, for students to see the connection between what they are learning and the real world because then they have a better understanding as to when they can apply what they have learned.  I felt that this was an important component to include so that students would see that mathematics is everywhere!  Although technology is not used often in this unit besides the use of a calculator, research shows that "many technologies function as scaffolds and tools to help students solve problems" (Bransford, Brown, Cocking, 2000, p. 211).  This is quite true because there is much time for scaffolding to occur when the students are exploring through problems on their own.  This is particularly true with the area of circles activity.  I feel that I have two other features in my unit that are unique.  Each day I share a math fact with the students and also present a problem of the day.  I feel that this helps motivate the students and grabs their attention.  When presenting these quotes, the students have the opportunity to see mathematics in a different light.  This is particularly true when telling a math joke.  This sheds a new light on mathematics for students, especially for students who may not like mathematics.  I think that it is important, also, to express a personal love for mathematics so that students can see and learn that mathematis is exciting and fun.  The teacher must express this, though, in order for the students to have a mathematical experience that may influence the rest of their mathematical career.  I also feel that this lesson tends to the multiple intelligences of students.  The seven intelligences include linguistic, logical, musical, spatial, bodily kinesthetic, interpersonal, and intrapersonal.  The "first two intelligences are those typically tapped on tests and most valued in schools"; thus, it is the teacher's job to incorporate the other intelligences into the classroom (
Bransford, et.al., 2000, p. 101).  I feel that this lesson incorporates activities that are appropriate for some of the different intelligences that students have. 

Brief Description of the Unit

This unit begins with an explanation of unit conversions for the customary and metric measurement systems.  This is a concept that is important for students to know in general, but it is also a concept that some of the components in the rest of the unit focus on.  The students will also learn about perimeter and circumference.  They use the information that they learned about the measurement systems in some of these mathematical explorations.  The students will also learn about how to find the area of parallelograms, triangles, trapezoids, and circles.  In addition to that, students are exposed to a day of exploration through problem solving and real world applications.  In this lesson, students are applying everything that they have learned and combining all of their mathematical knowledge together to figure out the problems.

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